Research in physical applied mathematics, motivated by real world
problems is central to my investigations. My research is generally in
the field of nonlinear waves with a focus on two areas: 1) fluid
dynamics of dispersive media with accompanying wave excitations
shock waves (DSWs) and solitary waves; 2) dynamics of ferromagnetic
media, spin torque, and localized excitations in nanomagnetism.
Methods employed include mathematical modeling, analysis, asymptotics,
Whitham modulation theory, and numerical analysis. Whenever possible,
comparisons with experiment are carried out.
The generation of DSWs represents a universal mechanism to resolve
hydrodynamic singularities in dispersive media. Physical
manifestations of DSWs include undular bores on shallow water and in
the atmosphere (the Morning Glory), nonlinear diffraction patterns in
optics, and matter waves in ultracold atoms. Any approximately
conservative, nonlinear, hydrodynamic medium exhibiting weak
dispersion can develop DSWs. The mathematical description of DSWs
involves a synthesis of methods from hyperbolic quasi-linear systems,
asymptotics, and soliton theory. This research
is supported by the National Science Foundation through grants
and CAREER DMS-1255422.
Ferromagnetic media provide a source of rich nonlinear, dispersive
phenomena with practical import. Theoretical and technological
developments have stimulated the field of nanomagnetism by the
introduction of spin polarized currents as a means to excite
magnetization dynamics at the nanometer scale in patterned
environments. Strongly nonlinear magnetic solitons were
observed in a nanomagnetic system. This solitary wave or
"droplet" joins the domain wall and magnetic vortex as a fundamental
and distinct object in nanomagnetism with similar potential for
fruitful science. This research is supported by the National Science
Foundation through a Career award, DMS-1255422.
A dynamic, visual representation of some of my scientific work is
exhibited in the linked pages below with animations of numerical simulations from some
problems that I have considered. Numerical simulations are excellent
for describing a particular problem with a particular set of
parameters. However, to understand the qualitative behavior, and to
be able to make informed predictions, mathematical analysis is
essential. The mathematical discussion has been kept to a minimum.
If you're interested in the math, read the papers! Not all of my
publications are represented here. To see a list,
click here. The animations can
be viewed with
Note that some of the simulation files can be somewhat large.